Question about lorentz-covariance of Dirac equation

[aiqun]

ψ(x) is the four-component wave function of the Dirac equation，that is ψ(x) can be expressed by a column vector (ψ1(x) ψ2(x) ψ3(x) ψ4(x)） ,under a lorentz transformation,it will become ψ'(x').I am confused that how ψ'(x') can be expressd
in the form which is stated by textbooks: ψ'(x')=S(a)ψ(x)
（a is the matrix of the lorentz transformation ,S(a) is a 4*4 matrix which is a function of the parameters of a ）
clearly.ψ'(x') is a funtion of x',ψ(x) is funtion of x,my question is for example ,how can
ψ'1(x') can be expressed by a linear combination of ψ1(x), ψ2(x) ,ψ3(x)and ψ4(x)?
is there someone can help me?

 

[malawi_glenn]

try just to do it for some special cases, e.g.

start from the dirac-plane waves at rest, and boost them to a momentum p

etc

 

[Avodyne]

First let's do a scalar field. A classical real scalar field assigns a real number to each point in spacetime. All observers agree on the value of that number for each point. If Alice uses coordinates x, she can write down a function $\varphi(x)$ that gives the number assigned by the field to the point she labels with coordinates x. Bob uses different coordinates x', related to Alice's by a Lorentz transformation, x'=ax. Bob also uses a different function, $\varphi'$, of his coordinates. However, since Bob and Alice agree on the value assigned by the field to any particular point, the numerical values of Bob's function of Bob's coordinates must agree with the numerical values of Alice's function of Alice's coordinates; that is, we must have $\varphi'(x')=\varphi(x)$.

For fields in other representations of the Lorentz group, such as a Dirac field, this gets generalized to $\psi'(x')=S(a)\psi(x)$, where S(a) is a matrix that acts on the index carried by the field.